Emin Karayel | Searcharxiv

arXiv2025

Derandomization with Pseudorandomness

Derandomization techniques are often used within advanced randomized algorithms. In particular, pseudorandom objects, such as hash families and expander graphs, are key components of such algorithms, but their verification presents a challenge. This work shows how such algorithms can be expressed and verified in Isabelle and presents a pseudorandom objects library that abstracts away the deep algebraic/analytic results involved. Moreover, it presents examples that show how the library eases and enables the verification of advanced randomized algorithms. Highlighting the value of this framework is that it was recently used to verify the space-optimal distinct elements algorithm by Blasiok from 2018, which relies on the combination of many derandomization techniques to achieve its optimality.

cs.LO↗

arXiv2023

An embarrassingly parallel optimal-space cardinality estimation algorithm

In 2020 Blasiok (ACM Trans. Algorithms 16(2) 3:1-3:28) constructed an optimal space streaming algorithm for the cardinality estimation problem with the space complexity of $\mathcal O(\varepsilon^{-2} \ln(δ^{-1}) + \ln n)$ where $\varepsilon$, $δ$ and $n$ denote the relative accuracy, failure probability and universe size, respectively. However, his solution requires the stream to be processed sequentially. On the other hand, there are algorithms that admit a merge operation; they can be used in a distributed setting, allowing parallel processing of sections of the stream, and are highly relevant for large-scale distributed applications. The best-known such algorithm, unfortunately, has a space complexity exceeding $Ω(\ln(δ^{-1}) (\varepsilon^{-2} \ln \ln n + \ln n))$. This work presents a new algorithm that improves on the solution by Blasiok, preserving its space complexity, but with the benefit that it admits such a merge operation, thus providing an optimal solution for the problem for both sequential and parallel applications. Orthogonally, the new algorithm also improves algorithmically on Blasiok's solution (even in the sequential setting) by reducing its implementation complexity and requiring fewer distinct pseudo-random objects.

cs.DS↗